"E" pronounced as "ye" after consonants will give you a strong accent (like pronouncing English "th" as "d" or "t"). So "нет" (no) does not sound as "nyet" ("ye" as in "yes"), it's more like "n'et" (soft "n", then "e" sound). Unfortunately, there's no way to spell it "correctly" in English.

Check out this video. In refrain (1:07) they repeat "Нет, нет, нет, нет, мы хотим сегодня. Нет, нет, нет, нет, мы хотим сейчас" ("No, No, No, We want (it) today. We want (it) now".
That's how "нет" (and stressed "e" after consonants) should sound.


Or more slowly here (0:47), and even slower at (1:12) Выхода нет (No way out):



In other words it's very important to learn to palatalize your consonants. It will help you with vowels pronunciation and vice versa.